Please complete the following prior to attending day 3 of the math academy:
1. Try the shirt & pant or cups, bowls & plates problem with your students and reflect on the lesson. Feel free to bring in student work to these problems.
2. How many towers three-high can be made from 2 colors? (Bring this to the next session, do not post here!)
3. Post your response to the following question:
We've had a chance to look at an overview of the Rutgers' long-term study. In what ways are the students in the study similar to your students? How are they different? (Don't forget to sign your name!)
My students had a great time participating in the shirt and pants activity. They really enjoyed the freedom of working something out for themselves, something I am sorry to say does not happen much in my grade. Different students approached it from different levels of understanding, and it was interesting for me to see their ways of thinking the problem through. Some students had an "aha" moment when they grasped the concept of attacking the problem in a logical manner. I would say they were similar to the students in the early grades of the study. Some shared ideas with others, while others just went about trying to solve it in their own way. In all, it was a successful lesson. I hope to allow them more freedom to think things through in the future.
Grace Clauss
My fifth graders are different because they all don't have the family support that the students in the study did. I'm sure just about all of us can say this. Parents just being there for the kids yields positive results (as we saw with the D'Andrea girl in the video). My students are similar because most do enjoy working in small groups and discussing ways to solve problems. They gain a great deal from those small group sessions. - Sarah Conway
My fifth grade students attacked these problems in the same way the students in the study did. They were very eager and had many different ways of solving the problem. They worked surprisingly well together. I had to laugh b/c one girl in my class commented on the outfit having to match like the girl in the video! I guess fashion can play a role in our problem solving! I did notice some of my students constantly wanting to know if they were correct or on the right track. I didn't remember seeing that in the video. The students seemed to not ask the teacher for help. Overall, I was pleased with the way my students worked together.- Debbie Putman
The students in our class appeared to be energetic and excited to participate in the activity. They worked individually and as partners. There was no one "running" the show. They cooperated and discussed their findings with each other.
Mary Ann Tranovich
Rebecca Giordano
Amy Seamans
(we all co-teach with each other, so we posted as one)
Posted 2 minutes ago
1-1 of 1
The students were placed into groups. Each group began devising strategies on how to tackle each problem. Two strategy words took form for both problems: "opposite" and "pattern." Those who followed the opposite strategy found themselves repeating arrangements often. Those who followed the pattern strategy were able to complete each task quicker and noticed their mistakes faster. The pattern groups were discussing and communicating ideas far more than the opposite groups. We tweaked the cube activity. The students used the cubes to make a 4 high tower. We hypothesized as to how many combinations there might be. We then took the idea of Blooms Taxonomy and extended the activity. The students went from a hands-on experience to using prior knowlegdge in making an educated guess on towers 3 high of two colors. They went on to draw and color the 3 high cube combinations without the concrete cubes. Most students were successful in this task. As in the Rutgers study, communication was very important. Those who communicated with eachother and bounced ideas off of others had greater success. The difference from the Rutgers study was there were some students who chose not to participate with others. Their success or lack of success actually revolved around their ability to cooperate with others.
Pam Ealo
Cheryl Decker
(co-teachers)
I did the activitiy with a small group of my students during homeroom time in the morning. I used a mixed group of students boy and girls and different levels. They were all interested in doing the activity at first, but I did find that one student in the group tended to "take charge" while the others then ended up agreeing with him. I am unsure if all really enjoyed it being clothing. I agree that it could be alteredto the interests of the group of students doing the activity.
Joann Tomassoni
My students are similiar to the students in the Rutgers study in that some of them enjoy explaining how they got to answer. However, some of my students do not like explaining how they came to answer. The often become frustrated, and sometimes shut down when pressed to explain themselves. My students did enjoy the activity with the clothing, but for the groups with boys I made it sports equipment. They worked well together, and were excited to figure out the answer. Some students did just sit back and passively agree with their peers, however the majority were working together.
-Kristy Kropa
Andrea James and Sharon Kinney, Mountain View Elementary , Co-Teachers 2nd grade: We put our students into groups of three, each were given 3 color crayons and drawing paper. Initially, they were unsure how to begin and were hesitant to talk to each other. With prompting, a sample conversation included "Do we draw the picture? Ans from us: Would that help? Students thought that when a shirt was used once it couldn't be used again. We encouraged them to mix and match. Once they got a few down on paper, Student 1, Student 2 and Student 3 started to compare their data to find common responses ( they circled or crossed out). They were the first to come up with 6 combinations - Student 1 is a learning support student and was the first to come up with a solution. This class is composed of at risk math students. We were happy with results and most groups were able to find 6 combinations.
I did the activity whole group at the second grade level. I first brought the students to the front of the room and explained to them the activity they were about to work on with a partner. I did the activity with the shirts and pants. I wrote on the chalkboard the items a person had. I named the person "John" so the students could make a connection of the clothes with a human. Once the students understood the items I continued to explain what their job was to do---finding as many combinations of outfits they could. They asked questions such as "Can we use two colors together", "Do we draw the people", "Can we use crayons to color the clothes?" and so on. My response was the same as another group, "Will that help you find the answer?" I let the students begin working as I walked around to observe. The top conversation with the groups was discussing outfits that "matched". I was surprised because the discussions were made by both girls and boys. I let the students work for about 30 minutes and then we came back together as a group to discuss the outcomes. Of the ten groups, only three came to the conclusion of 6 combinations. I found that the students didn't mind sharing their answers and the way they developed them. Their frustration came when working together because I didn't give them as much direction lead as I would in an ordinary lesson. I wanted to see how much they could solve without redirecting. Overall, the students did well. In their groups they talked about strategies to use and how to develop an answer together. There were some groups on the otherhand where one student did "take over" with ideas. ~Jessica Phillips~
My 5th grade students approached the cups, bowls, and plates lesson in a similar way. Some were very matter of fact in how they came to their answer, and I had to ask questions to bring out their thinking process (one of those was a gifted student who often solves math problems in her head). Others had a clear and organized plan of starting with a set color of an item and matching the other items to it until they found all the possibilities. They found that solving it that way, they could even have more than 10 combinations.Finally, those that didn't set out with an organized plan ended up searching through their combinations and then having to cross out doubles. I wonder if practicing more problems that require organized thinking will help our students to be more organized in daily life?!?
Sandy Willsch
My second grade students have been doing a math problem solving problem each morning during breakfast time. A couple of months ago they had the shirts/pants question. I decided to reuse it to see what they remembered and to get us started. The children worked in pairs and needed to be reminded to work together. So many wanted to do it themselves. Most students drew the shirts and pants and then the lines to show the six combinations. I was really pleased to see that they were willing to draw the model since often they just want to write down an answer. We then went on to the cups, bowls and plates. Many pairs used the same shirt/pant format which lead to a lot of lines. They figured that there were at least 10 lines and enough combinations. One pair though made a tally mark for each different combination which clarified their answer. A couple pairs of students made charts with ten columns. They wrote in each different combination. That was the easiest to see. Only a few didn't know what to do, but in time all pairs of students had an answer and a drawing or chart to prove it. They were engaged and worked well together once they were encouraged to talk to each other. I will definately do this again.
Elizabeth Haber
Jacqueline Ferenczi and Andrea James (Co-teachers) Mountain View Elementary, 4th grade. Students were placed in groups of 3 (one group had 4 students). Students were given drawing paper and crayons. The shirt/pants problem was posted on chart paper. The problem was posed to the students. As students began working in their groups, we observed the conversations and problem solving process of the students. Most students began to draw shirts and pants to replicate the problem at hand. That is, they drew yellow, blue and green shirts and then blue and green pants. Some of the groups, however, also drew yellow pants. After a clarification of the facts as given on the chart paper, they crossed out the combinations that included the yellow pants. Most interesting was a group of students who used tally marks rather than actual drawings of shirts and pants. The group was able to arrive at the correct answer of "6" in a short period of time. They used a method similar to that shown in the video. For example, they drew blue, green and yellow tallies for the shirts and below the shirts they drew blue and green tallies for the pants. They continued drawing lines connecting the remaining shirt and pant options. Most of the remaining groups also arrived at the correct answer. The groups shared their results and their strategies to solve the problem. This group activity went very well. It was fun watching the students work together.
I chose to do this activity with students in a class consisting of students in 7th and 8th grade. At first they moaned and groaned about having to work through a word problem (a little different from the Rutger's study where the students were eager to attack the problem), so we did an example together first. We used a mirror example of the "shirt & pant" problem, but we used ice cream and ice cream cones instead. They were so excited to be able to use white boards to draw their ice ceam and cones. After we oganized our thoughts and drew the pictures together, students had to come up with the different combinations on their own. Some students drew lines from the ice cream scoops to the cones, while others just shouted out possible combinations. Then there was one student who carefully drew 8 ice cream cones with ice ceam on top. It was great to see what they'd come up with on their own. We went over the results together, and I really think students enjoyed explaining how they developed their answers.
I then had students work through the shirt & pant problem on their own. Most students used pictures to represent the pants and the shirts. Two of my students wrote down the different outfit combinations. Then I had one student (one that I didn't even think was paying attention) who looked at the problem on the board and shouted out "there are 6 combinations". I quickly went to see how he'd come up with the correct response. He told me he multiplied the number of pant options (2) times the number of shirt options (3) to get 6 possible outcomes! I was really excited that he was able to make that connection on his own! So although my students didn't start off as eager as the students in the study, they really walked away with a feeling of accomplishment :) Tracy Suprick (Valley View Math Coach)
I chose to do the activities with my 9th grade Intro to Algebra class. They enjoyed working on the problem. Some of them got creative and wanted to know if they can layer shirts and if so how many. I thought this was funny, because I did the same thing at our last session. One of the early questions consisted of the words "Is this the right answer..." I guess it is just human nature. Looking at the student samples and explanations, I observed the similarities of my students (in their approach) to the students in the Rutgers study. Most used charts and tree diagrams. There was even two students worried about whether or not their opinion mattered in creating the outfits - much like Stephanie. However, I feel that some of my students differed in that their mathematical skills were more advanced than the students in the study as they knew how to find the answer simply by mulitiplying. They struggled when asked to explain "why 3(2) = 6). I think this was a great activity to do with my students - as I am always trying to stress the importance of "why" we do things in stead of just "how".
~ Ava Reynolds (Dunmore Math Coach)
The pants and shirts activity was done in my enrichment math group (verses an intervention/extra help group). The group consists of 22 third-graders. The children worked independently and quickly came up with their answers. I asked various students to share their work orally with brief discussions. Most had done the same process, but as others have noted, they still were asking, "is it right"? One thing that I think helped my students work so quickly was that I had drawn (with colored chalk) the shirts and pants on the board. I have found that many students, especially at this age, work better with some type of visual guidance in addition to the spoken or written word problem. Some students found the answer without much effort/thinking. Others needed to draw something on the paper. Some had letters for pants and shirts and then did some matching up. A reinforcement of multiple intelligences working and getting it right in the classroom. -Allison Gillis
I did the pants and shirts activity as well as the cups and bowls. The students worked in groups of 2. The children used a couple of different ways to get their answer and some had the answers wrong, just as the students in the video did. Some finished very quickly and used the basic counting principle as their explanation. They just did 3 x 2, while others drew themselves a diagram. Most students solved the problem just as the students in the video did. They used the diagram and drew lines or they used letters (B,W,Y) and made the various combinations. They had a much easier time doing the pants and shirts compared to the cups and bowls. Some students struggled with that many combinations while others once again used letters and combinations to solve. I only had about 2 students who used the basic counting principle of 3x3x2 to get the answer. It was fun to do in the classroom and the students really enjoyed the opportunity to solve the problems and be able to share their thinking with the class. I'd have to say my students handled and attacked the problem much the same as the students in the video. Overall it was a great experience and I hope to do more things like it in the future. - Kenny Hemmler (WW)
1. Try the shirt & pant or cups, bowls & plates problem with your students and reflect on the lesson. Feel free to bring in student work to these problems.
2. How many towers three-high can be made from 2 colors? (Bring this to the next session, do not post here!)
3. Post your response to the following question:
We've had a chance to look at an overview of the Rutgers' long-term study. In what ways are the students in the study similar to your students? How are they different? (Don't forget to sign your name!)
My students had a great time participating in the shirt and pants activity. They really enjoyed the freedom of working something out for themselves, something I am sorry to say does not happen much in my grade. Different students approached it from different levels of understanding, and it was interesting for me to see their ways of thinking the problem through. Some students had an "aha" moment when they grasped the concept of attacking the problem in a logical manner. I would say they were similar to the students in the early grades of the study. Some shared ideas with others, while others just went about trying to solve it in their own way. In all, it was a successful lesson. I hope to allow them more freedom to think things through in the future.
Grace Clauss
- My fifth graders are different because they all don't have the family support that the students in the study did. I'm sure just about all of us can say this. Parents just being there for the kids yields positive results (as we saw with the D'Andrea girl in the video). My students are similar because most do enjoy working in small groups and discussing ways to solve problems. They gain a great deal from those small group sessions. - Sarah Conway
- My fifth grade students attacked these problems in the same way the students in the study did. They were very eager and had many different ways of solving the problem. They worked surprisingly well together. I had to laugh b/c one girl in my class commented on the outfit having to match like the girl in the video! I guess fashion can play a role in our problem solving! I did notice some of my students constantly wanting to know if they were correct or on the right track. I didn't remember seeing that in the video. The students seemed to not ask the teacher for help. Overall, I was pleased with the way my students worked together.- Debbie Putman
- The students in our class appeared to be energetic and excited to participate in the activity. They worked individually and as partners. There was no one "running" the show. They cooperated and discussed their findings with each other.
Mary Ann TranovichRebecca Giordano
Amy Seamans
(we all co-teach with each other, so we posted as one)
Posted 2 minutes ago
1-1 of 1
The students were placed into groups. Each group began devising strategies on how to tackle each problem. Two strategy words took form for both problems: "opposite" and "pattern." Those who followed the opposite strategy found themselves repeating arrangements often. Those who followed the pattern strategy were able to complete each task quicker and noticed their mistakes faster. The pattern groups were discussing and communicating ideas far more than the opposite groups. We tweaked the cube activity. The students used the cubes to make a 4 high tower. We hypothesized as to how many combinations there might be. We then took the idea of Blooms Taxonomy and extended the activity. The students went from a hands-on experience to using prior knowlegdge in making an educated guess on towers 3 high of two colors. They went on to draw and color the 3 high cube combinations without the concrete cubes. Most students were successful in this task. As in the Rutgers study, communication was very important. Those who communicated with eachother and bounced ideas off of others had greater success. The difference from the Rutgers study was there were some students who chose not to participate with others. Their success or lack of success actually revolved around their ability to cooperate with others.
Pam Ealo
Cheryl Decker
(co-teachers)
I did the activitiy with a small group of my students during homeroom time in the morning. I used a mixed group of students boy and girls and different levels. They were all interested in doing the activity at first, but I did find that one student in the group tended to "take charge" while the others then ended up agreeing with him. I am unsure if all really enjoyed it being clothing. I agree that it could be alteredto the interests of the group of students doing the activity.
Joann Tomassoni
My students are similiar to the students in the Rutgers study in that some of them enjoy explaining how they got to answer. However, some of my students do not like explaining how they came to answer. The often become frustrated, and sometimes shut down when pressed to explain themselves. My students did enjoy the activity with the clothing, but for the groups with boys I made it sports equipment. They worked well together, and were excited to figure out the answer. Some students did just sit back and passively agree with their peers, however the majority were working together.
-Kristy Kropa
Andrea James and Sharon Kinney, Mountain View Elementary , Co-Teachers 2nd grade: We put our students into groups of three, each were given 3 color crayons and drawing paper. Initially, they were unsure how to begin and were hesitant to talk to each other. With prompting, a sample conversation included "Do we draw the picture? Ans from us: Would that help? Students thought that when a shirt was used once it couldn't be used again. We encouraged them to mix and match. Once they got a few down on paper, Student 1, Student 2 and Student 3 started to compare their data to find common responses ( they circled or crossed out). They were the first to come up with 6 combinations - Student 1 is a learning support student and was the first to come up with a solution. This class is composed of at risk math students. We were happy with results and most groups were able to find 6 combinations.
I did the activity whole group at the second grade level. I first brought the students to the front of the room and explained to them the activity they were about to work on with a partner. I did the activity with the shirts and pants. I wrote on the chalkboard the items a person had. I named the person "John" so the students could make a connection of the clothes with a human. Once the students understood the items I continued to explain what their job was to do---finding as many combinations of outfits they could. They asked questions such as "Can we use two colors together", "Do we draw the people", "Can we use crayons to color the clothes?" and so on. My response was the same as another group, "Will that help you find the answer?" I let the students begin working as I walked around to observe. The top conversation with the groups was discussing outfits that "matched". I was surprised because the discussions were made by both girls and boys. I let the students work for about 30 minutes and then we came back together as a group to discuss the outcomes. Of the ten groups, only three came to the conclusion of 6 combinations. I found that the students didn't mind sharing their answers and the way they developed them. Their frustration came when working together because I didn't give them as much direction lead as I would in an ordinary lesson. I wanted to see how much they could solve without redirecting. Overall, the students did well. In their groups they talked about strategies to use and how to develop an answer together. There were some groups on the otherhand where one student did "take over" with ideas. ~Jessica Phillips~
My 5th grade students approached the cups, bowls, and plates lesson in a similar way. Some were very matter of fact in how they came to their answer, and I had to ask questions to bring out their thinking process (one of those was a gifted student who often solves math problems in her head). Others had a clear and organized plan of starting with a set color of an item and matching the other items to it until they found all the possibilities. They found that solving it that way, they could even have more than 10 combinations.Finally, those that didn't set out with an organized plan ended up searching through their combinations and then having to cross out doubles. I wonder if practicing more problems that require organized thinking will help our students to be more organized in daily life?!?
Sandy Willsch
My second grade students have been doing a math problem solving problem each morning during breakfast time. A couple of months ago they had the shirts/pants question. I decided to reuse it to see what they remembered and to get us started. The children worked in pairs and needed to be reminded to work together. So many wanted to do it themselves. Most students drew the shirts and pants and then the lines to show the six combinations. I was really pleased to see that they were willing to draw the model since often they just want to write down an answer. We then went on to the cups, bowls and plates. Many pairs used the same shirt/pant format which lead to a lot of lines. They figured that there were at least 10 lines and enough combinations. One pair though made a tally mark for each different combination which clarified their answer. A couple pairs of students made charts with ten columns. They wrote in each different combination. That was the easiest to see. Only a few didn't know what to do, but in time all pairs of students had an answer and a drawing or chart to prove it. They were engaged and worked well together once they were encouraged to talk to each other. I will definately do this again.
Elizabeth Haber
Jacqueline Ferenczi and Andrea James (Co-teachers) Mountain View Elementary, 4th grade. Students were placed in groups of 3 (one group had 4 students). Students were given drawing paper and crayons. The shirt/pants problem was posted on chart paper. The problem was posed to the students. As students began working in their groups, we observed the conversations and problem solving process of the students. Most students began to draw shirts and pants to replicate the problem at hand. That is, they drew yellow, blue and green shirts and then blue and green pants. Some of the groups, however, also drew yellow pants. After a clarification of the facts as given on the chart paper, they crossed out the combinations that included the yellow pants. Most interesting was a group of students who used tally marks rather than actual drawings of shirts and pants. The group was able to arrive at the correct answer of "6" in a short period of time. They used a method similar to that shown in the video. For example, they drew blue, green and yellow tallies for the shirts and below the shirts they drew blue and green tallies for the pants. They continued drawing lines connecting the remaining shirt and pant options. Most of the remaining groups also arrived at the correct answer. The groups shared their results and their strategies to solve the problem. This group activity went very well. It was fun watching the students work together.
I chose to do this activity with students in a class consisting of students in 7th and 8th grade. At first they moaned and groaned about having to work through a word problem (a little different from the Rutger's study where the students were eager to attack the problem), so we did an example together first. We used a mirror example of the "shirt & pant" problem, but we used ice cream and ice cream cones instead. They were so excited to be able to use white boards to draw their ice ceam and cones. After we oganized our thoughts and drew the pictures together, students had to come up with the different combinations on their own. Some students drew lines from the ice cream scoops to the cones, while others just shouted out possible combinations. Then there was one student who carefully drew 8 ice cream cones with ice ceam on top. It was great to see what they'd come up with on their own. We went over the results together, and I really think students enjoyed explaining how they developed their answers.
I then had students work through the shirt & pant problem on their own. Most students used pictures to represent the pants and the shirts. Two of my students wrote down the different outfit combinations. Then I had one student (one that I didn't even think was paying attention) who looked at the problem on the board and shouted out "there are 6 combinations". I quickly went to see how he'd come up with the correct response. He told me he multiplied the number of pant options (2) times the number of shirt options (3) to get 6 possible outcomes! I was really excited that he was able to make that connection on his own! So although my students didn't start off as eager as the students in the study, they really walked away with a feeling of accomplishment :) Tracy Suprick (Valley View Math Coach)
I chose to do the activities with my 9th grade Intro to Algebra class. They enjoyed working on the problem. Some of them got creative and wanted to know if they can layer shirts and if so how many. I thought this was funny, because I did the same thing at our last session. One of the early questions consisted of the words "Is this the right answer..." I guess it is just human nature. Looking at the student samples and explanations, I observed the similarities of my students (in their approach) to the students in the Rutgers study. Most used charts and tree diagrams. There was even two students worried about whether or not their opinion mattered in creating the outfits - much like Stephanie. However, I feel that some of my students differed in that their mathematical skills were more advanced than the students in the study as they knew how to find the answer simply by mulitiplying. They struggled when asked to explain "why 3(2) = 6). I think this was a great activity to do with my students - as I am always trying to stress the importance of "why" we do things in stead of just "how".
~ Ava Reynolds (Dunmore Math Coach)
The pants and shirts activity was done in my enrichment math group (verses an intervention/extra help group). The group consists of 22 third-graders. The children worked independently and quickly came up with their answers. I asked various students to share their work orally with brief discussions. Most had done the same process, but as others have noted, they still were asking, "is it right"? One thing that I think helped my students work so quickly was that I had drawn (with colored chalk) the shirts and pants on the board. I have found that many students, especially at this age, work better with some type of visual guidance in addition to the spoken or written word problem. Some students found the answer without much effort/thinking. Others needed to draw something on the paper. Some had letters for pants and shirts and then did some matching up. A reinforcement of multiple intelligences working and getting it right in the classroom. -Allison Gillis
I did the pants and shirts activity as well as the cups and bowls. The students worked in groups of 2. The children used a couple of different ways to get their answer and some had the answers wrong, just as the students in the video did. Some finished very quickly and used the basic counting principle as their explanation. They just did 3 x 2, while others drew themselves a diagram. Most students solved the problem just as the students in the video did. They used the diagram and drew lines or they used letters (B,W,Y) and made the various combinations. They had a much easier time doing the pants and shirts compared to the cups and bowls. Some students struggled with that many combinations while others once again used letters and combinations to solve. I only had about 2 students who used the basic counting principle of 3x3x2 to get the answer. It was fun to do in the classroom and the students really enjoyed the opportunity to solve the problems and be able to share their thinking with the class. I'd have to say my students handled and attacked the problem much the same as the students in the video. Overall it was a great experience and I hope to do more things like it in the future. - Kenny Hemmler (WW)